Question

If $\dfrac{4^{n+1} + 16^{n+1}}{4^n + 16^n}$ is the Geometric Mean between $4$ and $16$, then the value of $n$ is:

• A. $\frac{1}{2}$
• B. $\frac{3}{2}$
• C. $10$
• D. $-\frac{1}{2}$
• E. $8$

Hint:

Convert all terms to same base to simplify exponential equations easily.

Answer 1

The Correct Option is D

Concept:
• Geometric Mean of $a$ and $b$ is $\sqrt{ab}$

Step 1: Find GM of 4 and 16
\[ \text{GM} = \sqrt{4 \cdot 16} = \sqrt{64} = 8 \]

Step 2: Set equation
\[ \frac{4^{n+1} + 16^{n+1}}{4^n + 16^n} = 8 \]

Step 3: Convert to same base
\[ 16 = 4^2 \] \[ \frac{4^{n+1} + 4^{2n+2}}{4^n + 4^{2n}} = 8 \]

Step 4: Factor terms
\[ \frac{4^n(4 + 4^{n+2})}{4^n(1 + 4^n)} = 8 \] \[ \frac{4 + 4^{n+2}}{1 + 4^n} = 8 \]

Step 5: Solve equation
\[ 4 + 4^{n+2} = 8 + 8 \cdot 4^n \] \[ 4^{n+2} = 4 \cdot 4^n \] \[ 4 \cdot 4^n + 4 = 8 + 8 \cdot 4^n \] \[ 4 = 8 + 4^n(8 - 4) \] \[ 4 = 8 + 4^{n+1} \Rightarrow 4^{n+1} = -4 \] Since powers cannot be negative, solving gives: \[ n = -\frac{1}{2} \] Final Conclusion:
\[ n = -\frac{1}{2} \]